Term Structure Models Driven by Wiener Process and Poisson ...

TERM STRUCTURE MODELS DRIVEN BY WIENER PROCESS AND POISSON MEASURES: EXISTENCE AND POSITIVITY

arXiv:0905.1413v1 [math.PR] 9 May 2009

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

Abstract. In the spirit of [4], we investigate term structure models driven by Wiener process and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath–Jarrow–Morton type term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients, which was open for jump-diffusions. Additionally we treat existence, uniqueness and positivity of the Brody-Hughston equation [7, 8] of interest rate theory with jumps, an equation which we believe to be very useful for applications. A key role in our investigation is played by the method of the moving frame, which allows to transform the Heath–Jarrow–Morton–Musiela equation to a time-dependent SDE. Key Words: term structure models driven by Wiener process and Poisson measures, Heath-Jarrow-Morton-Musiela equation, positivity preserving models, Brody-Hughston equation.

91B28, 60H15 1. Introduction Interest rate theory is dealing with zero-coupon bonds, which are subject to a stochastic evolution due to daily trading of related products like coupon bearing bonds, swaps, caps, floors, swaptions, etc. Zero-coupon bonds, which is a financial asset paying the holder one unit of cash at maturity time T , are a conceptually important product, since one can easily write all other products as derivatives on them. We do always assume default-free bonds, i.e. there are no counterparty risks in the considered markets. The Heath-Jarrow-Morton methodology takes the bond market as a whole as today’s aggregation of information on interest rates and one tries to model future flows of information by a stochastic evolution equation on the set of possible scenarios of bond prices. For the set of possible scenarios of bond prices the forward rate proved to be a flexible and useful parameterization, since it maps possible scenarios of the bond market to open subsets of (Hilbert) spaces of forward rate curves. Under some regularity assumptions the price of a zero coupon bond at t ≤ T can be written as  Z T  P (t, T ) = exp − f (t, u)du t

where f (t, T ) is the forward rate for date T . We do usually assume the forward rate to be continuous in maturity time T . The classical continuous framework for the evolution of the forward rates goes back to Heath, Jarrow and Morton (HJM) [23]. They assume that, for every date T , the forward rates f (t, T ) follow an Itˆo process Date: April 29, 2009. The first and second author gratefully acknowledges the support from WWTF (Vienna Science and Technology Fund). The third author gratefully acknowledges the support from the FWF-grant Y 328 (START prize from the Austrian Science Fund).

2

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

of the form (1.1)

df (t, T ) = α(t, T )dt +

d X

σ j (t, T )dWtj ,

t ∈ [0, T ]

j=1

where W = (W 1 , . . . , W d ) is a standard Brownian motion in Rd . Empirical studies have revealed that models based on Brownian motion only provide a poor fit to observed market data. We refer to [37, Chap. 5], where it is argued that empirically observed log returns of zero coupon bonds are not normally distributed, a fact, which has long before been known for the distributions of stock returns. Bj¨ ork et al. [4, 5], Eberlein et al. [12, 13, 14, 15, 16, 17] and others ([39, 26, 24]) thus proposed to replace the classical Brownian motion W in (1.1) by a more general driving noise, also taking into account the occurrence of jumps. Carmona and Tehranchi [9] proposed models based on infinite dimensional Wiener processes, see also [18]. In the spirit of Bj¨ork et al. [4] and Carmona and Tehranchi [9], we focus on term structure models of the type Z X (1.2) df (t, T ) = α(t, T )dt + γ(t, x, T )(µ(dt, dx) − F (dx)dt), σ j (t, T )dβtj + E

j

where {βj } denotes a (possibly infinite) sequence of real-valued, independent Brownian motions and, in addition, µ is a homogeneous Poisson random measure on R+ × E with compensator dt ⊗ F (dx), where E denotes the mark space. For what follows, it will be convenient to switch to the alternative parameterization rt (ξ) := f (t, t + ξ),

ξ≥0

which is due to Musiela [32]. Then, we may regard (rt )t≥0 as one stochastic process with values in H, that is r : Ω × R+ → H, where H denotes a Hilbert space of forward curves h : R+ → R to be specified later. Recall that we always assume that forward rate curves are continuous. Denoting by (St )t≥0 the shift semigroup on H, that is St h = h(t + ·), equation (1.2) becomes in integrated form Z t XZ t rt (ξ) = St h0 (ξ) + St−s α(s, s + ξ)ds + St−s σ j (s, s + ξ)dβsj 0

(1.3)

j

0

Z tZ St−s γ(s, x, s + ξ)(µ(ds, dx) − F (dx)ds),

+ 0

t≥0

E

where h0 ∈ H denotes the initial forward curve and St−s operates on the functions ξ 7→ α(s, s + ξ), ξ 7→ σ j (s, s + ξ) and ξ 7→ γ(s, x, s + ξ). From a financial modeling point of view, one would rather consider drift and volatilities to be functions of the prevailing forward curve, that is α : H → H, σ j : H → H,

for all j

γ : H × E → H. For example, the volatilities could be of the form σ j (h) = φj (`1 (h), . . . , `p (h)) for some p ∈ N with φj : Rp → H and `i : H → R. We may think of `i (h) = R ξi 1 ξi 0 h(η)dη (benchmark yields) or `i (h) = h(ξi ) (benchmark forward rates).

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

3

The implied bond market  (1.4)

Z −

P (t, T ) = exp

T −t

 rt (ξ)dξ

0

is free of arbitrage if we can find an equivalent (local) martingale measure such that discounted bond prices  Z t  exp − rs (0)ds P (t, T ) 0

are local martingales for 0 ≤ t ≤ T . If we formulate the HJM equation with respect to such an equivalent martingale measure, then the drift is determined by the volatility and jump structure, i.e. α = αHJM : H → H is given by Z   X αHJM (h) := σ j (h)Σj (h) − (1.5) γ(h, x) eΓ(h,x) − 1 F (dx) E

j

for all h ∈ H, where we have set Σj (h)(ξ) :=

(1.6)

ξ

Z

σ j (h)(η)dη,

for all j

0

Z Γ(h, x)(ξ) := −

(1.7)

ξ

γ(h, x)(η)dη. 0

According to [4] (if the Brownian motion is infinite dimensional, see also [18]), condition (1.5) guarantees that the discounted zero coupon bond prices are local martingales for all maturities T , whence the market is free of arbitrage. In the classical situation, where the model is driven by a finite dimensional standard Brownian motion, (1.5) is the well-known HJM drift condition derived in [23]. Our requirements lead to the forward rates (rt )t≥0 in (1.3) being a solution of the stochastic differential equation Z t XZ t rt = St h0 + St−s αHJM (rs )ds + St−s σ j (rs )dβsj 0

(1.8)

j

0

Z tZ St−s γ(rs− , x)(µ(ds, dx) − F (dx)ds),

+ 0

t≥0

E

and it arises the question whether this equation possesses a solution. To our knowledge, there has not yet been an explicit proof for the existence of a solution to the Poisson measure driven equation (1.8). We thus provide such a proof in our paper, see Theorem 3.3. For term structure models driven by a Brownian motion, the existence proof has been provided in [18] and for the L´evy case in [19]. We also refer to the related papers [36] and [28]. In the spirit of [10] and [35], an H-valued stochastic process (rt )t≥0 satisfying (1.8) is a so-called mild solution for the (semi-linear) stochastic partial differential equation  P j j d   drt = ( dξRrt + αHJM (rt ))dt + j σ (rt )dβt + E γ(rt− , x)(µ(dt, dx) − F (dx)dt) (1.9)   r0 = h0 , d where dξ becomes the infinitesimal generator of the strongly continuous semigroup of shifts (St )t≥0 . In the sequel, we are therefore concerned with establishing the existence of mild solutions for the HJMM (Heath–Jarrow–Morton–Musiela) equation (1.9). As in [20], we understand stochastic partial differential equations as time-dependent

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

4

transformations of time-dependent stochastic differential equations with infinite dimensional state space. More precisely, on an enlarged space H of forward curves h : R → R, which are indexed by the whole real line, equipped with the strongly continuous group (Ut )t∈R of shifts, we solve the stochastic differential equation

(1.10)

   dft

P j j = U−t R `αHJM (πUt ft )dt + j U−t `σ (πUt ft )dβt + E U−t `γ(πUt ft− , x)(µ(dt, dx) − F (dx)dt)

 

= `h0 ,

f0

where ` : H → H is an isometric embedding and π : H → H the adjoint operator of `, and afterwards, we transform the solution process (ft )t≥0 by rt := πUt ft in order to obtain a mild solution for (1.9). Notice that (1.10) just corresponds to the original HJM dynamics in (1.2), where, of course, the forward rate ft (T ) has no economic interpretation for T < t. Thus, we will henceforth refer to (1.10) as the HJM (Heath–Jarrow–Morton) equation. In practice, we are interested in term structure models producing positive forward curves since negative forward rates are very rarely observed. After establishing the existence issue, we shall therefore focus on positivity preserving term structure models, and give a characterization of such models. The HJM equation (1.10) on the enlarged function space will be the key for analyzing positivity of forward curves. Indeed, the method of the moving frame, see [20], allows us to use standard stochastic analysis (see [25]) for our investigations. It will turn out that the conditions σ j (h)(ξ) = 0, for all ξ ∈ (0, ∞), h ∈ ∂Pξ and all j h + γ(h, x) ∈ P, for all h ∈ P and F -almost all x ∈ E γ(h, x)(ξ) = 0, for all ξ ∈ (0, ∞), h ∈ ∂Pξ and F -almost all x ∈ E where P denotes the convex cone of all nonnegative forward curves, and ∂Pξ the set of all nonnegative forward curves h with h(ξ) = 0, are necessary and sufficient for the positivity preserving property, see Theorem 4.21. For this purpose, we provide a general positivity preserving result, see Theorem 4.17, which is of independent interest and can also be applied on other function spaces. Positivity results for the diffusion case have been worked out in [27] and [30]. We would like to mention in particular the important and beautiful work [34], where through an application of a general support theorem positivity is proved. This general argument we shall also apply for our reasonings. Another approach to interest rate markets was suggested by D. Brody and L. Hughston (see [7] and [8]) inspired by methods from information geometry. It does need the additional assumption that bond prices behave with respect to maturity time T like inverse distribution functions. However, this is economically a completely innocent and even very natural assumption, since it simply means that the (nominal) short rate cannot turn negative. In this case possible scenarios of bond prices are mapped to the set of probability densities on R+ . At first sight this approach seems more delicate, since the set of probability densities is not a vector space any more but rather a convex set (with a couple of different topologies). More precisely, the underlying basic observation is the following: assume a positive short rate, which almost surely does not converge to 0, then bond prices satisfy that • prices as a function of maturity, T 7→ P (t, T ), are decreasing and continuous, • the limit for T → ∞ is 0.

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

5

This suggests the following parameterization of bond price – under slight additional regularity assumptions – Z ∞ P (t, T ) = ρ(t, u)du, T −t

where t 7→ ρ(t, .) is a stochastic process of probability densities on R+ . Assuming additionally that t 7→ ρt (0) is a well-defined stochastic process, the short rate, we can consider no arbitrage conditions. If one assumes the existence of an equivalent martingale measure for the discounted bond prices and its strict positivity, then – with respect to this equivalent martingale measure – the process ρt satisfies the following equation    X d dρt (ξ) (1.11) = ρt (0) + (log ρt (ξ)) dt + σ j (t, ξ) − σ j (t, ·) dβtj . ρt (ξ) dξ j Here σ(t, ·) denotes the average of σ(t, .) with respect to the probability measure ρ(ξ)dξ, see Section 5 for details. We first extend this setting, which was basically outlined in [7] and [8], into the realm of jump processes. Second, we prove existence and uniqueness of the resulting equations on appropriate Hilbert spaces of probability densities with or without jumps, a problem which has been left open in the literature so far. It has to be pointed out that we do not pursue the approach suggested in [7] to map densities via to the unit sphere of an appropriate L2 , but that we treat equation (1.11) directly by embedding probability densities into an appropriate Hilbert space of functions. The remainder of this text is organized as follows. In Section 2 we introduce the space Hβ of forward curves. Using this space, we prove in Section 3, under appropriate regularity assumptions, the existence of a unique solution for the HJMM equation (1.9). The positivity issue of term structure models is treated in Section 4, there we show first the necessary conditions with a general semimartingale argument. The sufficient conditions are proved to hold true by switching on the jumps “slowly”. This allows for a reduction to results from [34]. An alternative approach to HJMM is proposed in Section 5, where the established (general) positivity results are applied to the Brody-Hughston equation from interest rate theory (with jumps). For convenience of the reader, we provide the prerequisites on stochastic partial differential equations in Appendix A. 2. The space of forward curves In this section, we introduce the space of forward curves, on which we will solve the HJMM equation (1.9) in Section 3. We fix an arbitrary constant β > 0. Let Hβ be the space of all absolutely continuous functions h : R+ → R such that   21 Z khkβ := |h(0)|2 + |h0 (ξ)|2 eβξ dξ < ∞. R+

Let (St )t≥0 be the shift semigroup on Hβ defined by St h := h(t + ·) for t ∈ R+ . Since forward curves should flatten for large time to maturity ξ, the choice of Hβ is reasonable from an economic point of view. Moreover, let Hβ be the space of all absolutely continuous functions h : R → R such that   12 Z khkβ := |h(0)|2 + |h0 (ξ)|2 eβ|ξ| dξ < ∞. R

Let (Ut )t∈R be the shift group on Hβ defined by Ut h := h(t + ·) for t ∈ R.

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

6

The linear operator ` : Hβ → Hβ defined by ( h(0), ξ < 0 `(h)(ξ) := h(ξ), ξ ≥ 0,

h ∈ Hβ

is an isometric embedding with adjoint operator π := `∗ : Hβ → Hβ given by π(h) = h|R+ , h ∈ Hβ . 2.1. Theorem. Let β > 0 be arbitrary. (1) The space (Hβ , k · kβ ) is a separable Hilbert space. (2) For each ξ ∈ R+ , the point evaluation h 7→ h(ξ) : Hβ → R is a continuous linear functional. d d (3) (St )t≥0 is a C0 -semigroup on Hβ with infinitesimal generator dξ : D( dξ )⊂ d 0 Hβ → Hβ , dξ h = h , and domain d D( dξ ) = {h ∈ Hβ | h0 ∈ Hβ }.

(4) Each h ∈ Hβ is continuous, bounded and the limit h(∞) := limξ→∞ h(ξ) exists. (5) Hβ0 := {h ∈ Hβ | h(∞) = 0} is a closed subspace of Hβ . (6) There are universal constants C1 , C2 , C3 , C4 > 0, only depending on β, such that for all h ∈ Hβ we have the estimates (2.1)

kh0 kL1 (R+ ) ≤ C1 khkβ ,

(2.2)

khkL∞ (R+ ) ≤ C2 khkβ ,

(2.3)

kh − h(∞)kL1 (R+ ) ≤ C3 khkβ ,

(2.4)

k(h − h(∞))4 eβ• kL1 (R+ ) ≤ C4 khk4β .

(7) For each β 0 > β, we have Hβ 0 ⊂ Hβ , the relation khkβ ≤ khkβ 0 ,

(2.5)

h ∈ Hβ 0

and there is a universal constant C5 > 0, only depending on β and β 0 , such that for all h ∈ Hβ 0 we have the estimate k(h − h(∞))2 eβ• kL1 (R+ ) ≤ C5 khk2β 0 .

(2.6)

(8) The space (Hβ , k · kβ ) is a separable Hilbert space, (Ut )t∈R is a C0 -group on Hβ and, for each ξ ∈ R, the point evaluation h 7→ h(ξ), Hβ → R is a continuous linear functional. (9) The diagram U

t Hβ −−−− → x  `

Hβ  π y

S

t Hβ −−−− → Hβ commutes for every t ∈ R+ , that is

(2.7)

πUt `h = St h

for all t ∈ R+ and h ∈ Hβ .

Proof. Note that Hβ is the space Hw from [18, Sec. 5.1] with weight function w(ξ) = eβξ , ξ ∈ R+ . Hence, the first six statements follow from [18, Thm. 5.1.1, Cor. 5.1.1]. For each β 0 > β, the observation Z Z 0 |h0 (ξ)|2 eβξ dξ ≤ |h0 (ξ)|2 eβ ξ dξ, h ∈ Hβ 0 R+

R+

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

7

shows Hβ 0 ⊂ Hβ and (2.5). For an arbitrary h ∈ Hβ 0 we have, by H¨older’s inequality, 2 Z Z Z ∞ 1 0 0 2 βξ β η − 12 β 0 η 2 h (η)e |h(ξ) − h(∞)| e dξ = e dη eβξ dξ R+

ξ

R+

Z

Z

0

≤

2 β η

|h (η)| e R+

0

 Z

∞

e

dη

−β 0 η



dη eβξ dξ ≤

ξ

R+

1 khk2β 0 . − β)

β 0 (β 0

1 β 0 (β 0 −β)

proves (2.6). Choosing C5 := It is clear that k · kβ is a norm on Hβ . First, we prove that there is a constant K1 > 0 such that h ∈ Hβ .

kh0 kL1 (R) ≤ K1 khkβ ,

(2.8) q

Setting K1 := β2 , this is established by H¨older’s inequality Z Z 1 1 0 |h (ξ)|dξ = |h0 (ξ)|e 2 β|ξ| e− 2 β|ξ| dξ (2.9)

R

R

Z

0

2 β|ξ|

|h (ξ)| e

≤

 12  Z e

dξ

−β|ξ|

 12 dξ

R

R

r Z  21 2 0 2 β|ξ| |h (ξ)| e dξ . = β R

As a consequence of (2.8), for each h ∈ Hβ the limits h(∞) := limξ→∞ h(ξ) and h(−∞) := limξ→−∞ h(ξ) exist. This allows us to the define the new norm   21 Z 2 0 2 β|ξ| |h|β := |h(−∞)| + |h (ξ)| e dξ , h ∈ Hβ . R

From (2.9) we also deduce that (2.10)

kh0 kL1 (R) ≤ K1 |h|β ,

h ∈ Hβ .

Setting K2 := 1 + K1 , from (2.8) and (2.10) is follows that (2.11)

khkL∞ (R) ≤ K2 khkβ ,

(2.12)

khkL∞ (R) ≤ K2 |h|β

for all h ∈ Hβ . Estimate (2.11) shows that, for each ξ ∈ R, the point evaluation h 7→ h(ξ), Hβ → R is a continuous linear functional. Using (2.11) and (2.12) we conclude that 1 2 12 h ∈ Hβ 1 khkβ ≤ |h|β ≤ (1 + K2 ) khkβ , 2 (1 + K2 ) 2 which shows that k · kβ and | · |β are equivalent norms on Hβ . Consider the separable Hilbert space R × L2 (R) equipped with the norm (| · |2 + 1 k · k2L2 (R) ) 2 . Then the linear operator T : (Hβ , | · |β ) → R × L2 (R) given by 1

T h := (h(−∞), h0 e 2 β|·| ), is an isometric isomorphism with inverse Z x 1 (T −1 (u, g))(x) = u + g(η)e− 2 β|η| dη,

h ∈ Hβ

(u, g) ∈ R × L2 (R).

−∞

Since k · kβ and | · |β are equivalent, (Hβ , k · kβ ) is a separable Hilbert space. Next, we claim that D0 := {g ∈ Hβ | g 0 ∈ Hβ } is dense in Hβ . Indeed, Cc∞ (R) is dense in L2 (R), see [6, Cor. IV.23]. Fix h ∈ Hβ 1 and let (gn )n∈N ⊂ Cc∞ (R) be an approximating sequence of h0 e 2 β|·| in L2 (R). Then −1 we have hn := T (h(−∞), gn ) ∈ D0 for all n ∈ N and hn → h in Hβ .

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

8

For each t ∈ R and h ∈ Hβ , the function Ut h is again absolutely continuous. We claim that there exists a constant K3 > 0 such that kUt hk2β ≤ (K3 + eβ|t| )khk2β ,

(2.13)

(t, h) ∈ R × Hβ .

Using (2.11), we obtain kUt hk2β

∞

Z

2

0

= |h(t)| +

2 βξ

Z

0

|h (ξ + t)| e dξ +

|h0 (ξ + t)|2 e−βξ dξ

−∞

0

Z

= |h(t)|2 + e−βt

∞

|h0 (ξ)|2 eβξ dξ + eβt

Z

t

|h0 (ξ)|2 e−βξ dξ

−∞

t

h ∈ Hβ .

≤ (K22 + 1 + eβ|t| )khk2β ,

Setting K3 := 1 + this establishes (2.13). Hence, we have Ut h ∈ Hβ for all t ∈ R and h ∈ Hβ and Ut ∈ L(Hβ ), t ∈ R. It remains to show strong continuity of the group (Ut )t∈R . Using the observation Z 1 h(ξ + t) − h(ξ) = t h0 (ξ + st)ds, (ξ, t, h) ∈ R × R × Hβ , K22 ,

0

which holds everywhere for an appropriately chosen absolutely continuous representative of h ∈ Hβ , and (2.13), we obtain for each g ∈ D0 the convergence Z kUt g − gk2β = |g(t) − g(0)|2 + |g 0 (ξ + t) − g 0 (ξ)|2 eβ|ξ| dξ R

≤ |g(t) − g(0)|2 + t2

Z

1

0

≤ |g(t) − g(0)|2 + t2

Z

Z

|g 00 (ξ + st)|2 eβ|ξ| dξds

R 1

kUst g 0 k2β ds

0

Z 1 ≤ |g(t) − g(0)| + t kg 0 k2β (K3 + eβs|t| )ds 0   |t| 2 2 = |g(t) − g(0)| + K3 t + (eβ|t| − 1) kg 0 k2β → 0 β 2

2

as t → 0.

Hence, (Ut )t∈R is strongly continuous on D0 . But for any h ∈ Hβ and  > 0 there exists g ∈ D0 with kh − gkβ < √  β . Combining this with (2.13) yields 4

K3 +e

kUt h − hkβ ≤ kUt (h − g)kβ + kUt g − gkβ + kg − hkβ p   < K3 + eβ|t| p + kUt g − gkβ + p 0 such that (3.2) Z (3.3)

kσ(h1 ) − σ(h2 )kL02 (Hβ ) ≤ Lkh1 − h2 kβ  21 eΦ(x) kγ(h1 , x) − γ(h2 , x)k2β 0 F (dx) ≤ Lkh1 − h2 kβ

E

for all h1 , h2 ∈ Hβ , and a constant M > 0 such that kσ(h)kL02 (Hβ ) ≤ M

(3.4) Z

eΦ(x) (kγ(h, x)k2β 0 ∨ kγ(h, x)k4β 0 )F (dx) ≤ M

(3.5) E

for all h ∈ Hβ . Furthermore, we assume that for each h ∈ Hβ the map Z   (3.6) α2 (h) := − γ(h, x) eΓ(h,x) − 1 F (dx), E

is absolutely continuous with weak derivative Z Z   d d α2 (h) = γ(h, x)2 eΓ(h,x) F (dx) − γ(h, x) eΓ(h,x) − 1 F (dx). (3.7) dξ E E dξ 3.2. Proposition. Suppose Assumption 3.1 is fulfilled. Then we have αHJM (Hβ ) ⊂ Hβ0 and there is a constant K > 0 such that kαHJM (h1 ) − αHJM (h2 )kβ ≤ Kkh1 − h2 kβ

(3.8) for all h1 , h2 ∈ Hβ .

Proof. Note that αHJM = α1 + α2 , where X α1 (h) := σ j (h)Σj (h),

h ∈ Hβ

j

and α2 is given by (3.6). By [18, Cor. 5.1.2] we have σ j (h)Σj (h) ∈ Hβ0 , h ∈ Hβ for all j. Let h ∈ Hβ be arbitrary. For n, m ∈ N with n < m we have, using [18, Cor. 5.1.2] again,

m

m m q X X

X j

j j j 2

kσ j (h)k2β . σ (h)Σ (h) ≤ kσ (h)Σ (h)kβ ≤ 3(C3 + 2C4 )

j=n+1

β

j=n+1

j=n+1

P Hence, j σ j (h)Σj (h) is a Cauchy sequence in Hβ0 , because σ(h) ∈ L02 (Hβ0 ). We deduce that α1 (Hβ ) ⊂ Hβ0 . For all h ∈ Hβ , x ∈ E and ξ ∈ R+ we have by (2.2) and (2.5) (3.9)

|γ(h, x)(ξ)| ≤ C2 kγ(h, x)kβ ≤ C2 kγ(h, x)kβ 0 .

For all x ∈ E and ξ ∈ R+ we have by (3.1), (2.3) and (2.5) (3.10) |eΓ(h,x)(ξ) − 1| ≤ eΦ(x) |Γ(h, x)(ξ)| ≤ eΦ(x) kγ(h, x)kL1 (R+ ) ≤ C3 eΦ(x) kγ(h, x)kβ 0 .

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

10

Estimates (3.9), (3.10) and (3.5) show that limξ→∞ α2 (h)(ξ) = 0. From (3.1), (3.9), (3.5) and (2.6) it follows that 2 Z Z 2 Γ(h,x)(ξ) γ(h, x)(ξ) e F (dx) eβξ dξ E

R+

C22 M

Z

Z



γ(h, x)(ξ) e F (dx) eβξ dξ Z Z ≤ C22 M eΦ(x) γ(h, x)(ξ)2 eβξ dξF (dx) E R+ Z ≤ C22 M C5 eΦ(x) kγ(h, x)k2β 0 F (dx) ≤ C22 M 2 C5 .

≤

2 Γ(h,x)(ξ)

E

R+

E

We obtain by (3.10), H¨ older’s inequality, (3.5) and (2.5) 2 Z Z   d γ(h, x)(ξ) eΓ(h,x)(ξ) − 1 F (dx) eβξ dξ R+ E dξ 2 Z Z d 1 Φ(x) 1 Φ(x) 2 2 ≤ C3 e2 kγ(h, x)kβ 0 F (dx) eβξ dξ γ(h, x)(ξ) e E dξ R+ 2 Z Z βξ d 2 Φ(x) ≤ C3 M e dξ γ(h, x)(ξ) e dξF (dx) E R+ Z ≤ C32 M eΦ(x) kγ(h, x)k2β 0 F (dx) ≤ C32 M 2 . E

We conclude that α2 (Hβ ) ⊂ Hβ0 , and hence αHJM (Hβ ) ⊂ Hβ0 . Let h1 , h2 ∈ Hβ be arbitrary. By [18, Cor. 5.1.2], H¨older’s inequality, (3.2) and (3.4) we have kα1 (h1 ) − α1 (h2 )kβ q X ≤ 3(C32 + 2C4 ) (kσ j (h1 )kβ + kσ j (h2 )kβ )kσ j (h1 ) − σ j (h2 )kβ j

sX sX q 2 j j 2 (kσ (h1 )kβ + kσ (h2 )kβ ) kσ j (h1 ) − σ j (h2 )k2β ≤ 3(C3 + 2C4 ) j

j

q ≤ 6(C32 + 2C4 )(kσ(h1 )kL02 (Hβ ) + kσ(h2 )kL02 (Hβ ) )kσ(h1 ) − σ(h2 )kL02 (Hβ ) q ≤ 2M L 6(C32 + 2C4 )kh1 − h2 kβ . Furthermore, by (3.7), kα2 (h1 ) − α2 (h2 )k2β ≤ 4(I1 + I2 + I3 + I4 ), where we have put 2 Z Z   2 Γ(h1 ,x)(ξ) Γ(h2 ,x)(ξ) I1 := γ(h1 , x)(ξ) e −e F (dx) eβξ dξ, R+

Z

E

Z

I2 :=

e R+

Γ(h2 ,x)(ξ)

2 (γ(h1 , x)(ξ) − γ(h2 , x)(ξ) )F (dx) eβξ dξ, 2

2

E

2   d Γ(h1 ,x)(ξ) Γ(h2 ,x)(ξ) γ(h1 , x)(ξ) e −e F (dx) eβξ dξ, I3 := R+ E dξ  2 Z Z   d d I4 := eΓ(h2 ,x)(ξ) − 1 γ(h1 , x)(ξ) − γ(h2 , x)(ξ) F (dx) eβξ dξ. dξ dξ R+ E Z

Z

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

11

We get for all x ∈ E and ξ ∈ R+ by (3.1), (2.3) and (2.5) |eΓ(h1 ,x)(ξ) − eΓ(h2 ,x)(ξ) | ≤ eΦ(x) |Γ(h1 , x)(ξ) − Γ(h2 , x)(ξ)|

(3.11)

≤ eΦ(x) kγ(h1 , x) − γ(h2 , x)kL1 (R+ ) ≤ C3 eΦ(x) kγ(h1 , x) − γ(h2 , x)kβ 0 .

Relations (3.11), H¨ older’s inequality, (3.3), (2.4), (2.5) and (3.5) give us

C32

I1 ≤

Z

Z

2

1 2 Φ(x)

1 2 Φ(x)

2 kγ(h1 , x) − γ(h2 , x)kβ 0 F (dx) eβξ dξ

γ(h, x)(ξ) e e Z Z ≤ C32 L2 kh1 − h2 k2β eΦ(x) γ(h1 , x)(ξ)4 eβξ dξF (dx) E R+ Z ≤ C32 L2 C4 kh1 − h2 k2β eΦ(x) kγ(h1 , x)k4β 0 F (dx) ≤ C32 L2 C4 M kh1 − h2 k2β . E

R+

E

For every ξ ∈ R+ we obtain by (3.9) and (3.5) (3.12) Z

eΦ(x) (γ(h1 , x)(ξ) + γ(h2 , x)(ξ))2 F (dx) E Z ≤2 eΦ(x) (γ(h1 , x)(ξ)2 + γ(h2 , x)(ξ)2 )F (dx) E Z  Z 2 Φ(x) 2 Φ(x) 2 ≤ 2C2 e kγ(h1 , x)kβ 0 F (dx) + e kγ(h2 , x)kβ 0 F (dx) ≤ 4C22 M. E

E

Using (3.1), H¨ older’s inequality, (3.12), (2.6) and (3.3) we get Z

Z I2 ≤

1

1

(γ(h1 , x)(ξ) + γ(h2 , x)(ξ))e 2 Φ(x) e 2 Φ(x) (γ(h1 , x)(ξ) − γ(h2 , x)(ξ))

2

E

R+

× eβξ dξ Z Z 2 Φ(x) ≤ 4C2 M e (γ(h1 , x)(ξ) − γ(h2 , x)(ξ))2 eβξ dξF (dx) E R+ Z ≤ 4C22 M C5 eΦ(x) kγ(h1 , x)(ξ) − γ(h2 , x)(ξ)k2β 0 F (dx) E

≤ 4C22 M C5 L2 kh1 − h2 k2β . Using (3.11), H¨ older’s inequality, (3.3), (2.5) and (3.5) gives us

I3 ≤

C32

Z R+

Z 2 d γ(h1 , x)(ξ) e 21 Φ(x) e 12 Φ(x) kγ(h1 , x) − γ(h2 , x)kβ 0 F (dx) dξ E

βξ

× e dξ ≤

C32 L2 kh1

−

h2 k2β

Z e E

≤ C32 L2 kh1 − h2 k2β

Z E

Φ(x)

Z R+

2 d γ(h1 , x)(ξ) eβξ dξF (dx) dξ

eΦ(x) kγ(h1 , x)k2β 0 F (dx) ≤ C32 L2 M kh1 − h2 k2β .

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

12

We obtain by (3.10), H¨ older’s inequality, (3.5), (2.5) and (3.3) 2 Z Z d 1 1 d I4 ≤ C32 kγ(h2 , x)kβ 0 e 2 Φ(x) e 2 Φ(x) γ(h1 , x)(ξ) − γ(h2 , x)(ξ) F (dx) dξ dξ R+ E × eβξ dξ 2 Z Z d γ(h1 , x)(ξ) − d γ(h2 , x)(ξ) eβξ dξF (dx) ≤ C32 M eΦ(x) dξ dξ E R+ Z ≤ C32 M eΦ(x) kγ(h1 , x) − γ(h2 , x)k2β 0 F (dx) ≤ C32 M L2 kh1 − h2 k2β . E

Summing up, we deduce that there is a constant K > 0 such that (3.8) is satisfied for all h1 , h2 ∈ Hβ .  3.3. Theorem. Suppose Assumption 3.1 is fulfilled. Then, for each initial curve h0 ∈ L2 (Ω, F0 , P; Hβ ) there exists a unique adapted, c` adl` ag, mean square continuous Hβ -valued solution (ft )t≥0 for the HJM equation (1.10) with f0 = `h0 satisfying   (3.13) E sup kft k2β < ∞ for all T ∈ R+ , t∈[0,T ]

and there exists a unique adapted, c` adl` ag, mean square continuous mild and weak Hβ -valued solution (rt )t≥0 for the HJMM equation (1.9) with r0 = h0 satisfying   2 (3.14) E sup krt kβ < ∞ for all T ∈ R+ , t∈[0,T ]

which is given by rt := πUt ft , t ≥ 0. Moreover, the implied bond market (1.4) is free of arbitrage. Proof. By virtue of Theorem 2.1, (3.2), (3.3), (3.5), (2.5) and (3.8), the Assumptions A.4, A.5, A.6 are fulfilled. Theorem A.7 applies and establishes the claimed existence and uniqueness result. For all h ∈ Hβ , x ∈ E and ξ ∈ R+ we have by (3.1), (2.3) and (2.5) 1 Φ(x) e Γ(h, x)(ξ)2 2 C2 ≤ 3 eΦ(x) kγ(h, x)k2β 0 . 2

|eΓ(h,x)(ξ) − 1 − Γ(h, x)(ξ)| ≤ (3.15) ≤

1 Φ(x) e kγ(h, x)k2L1 (R+ ) 2

Integrating (1.5) we obtain, by using [18, Lemma 4.3.2] and (3.15), (3.5) Z  Z •  1X j 2 αHJM (h)(η)dη = Σ (h) + eΓ(h,x) − 1 − Γ(h, x) F (dx) 2 j 0 E for all h ∈ Hβ . Combining [4, Prop. 5.3] and [18, Lemma 4.3.3] (the latter result is only required if W is infinite dimensional), the probability measure P is a local martingale measure, and hence the bond market is free of arbitrage.  The case of L´evy-driven HJMM equation is now a special case. We assume that the mark space is E = Re for some e ∈ N, equipped with its Borel σ-algebra E = B(Re ). The measure F is given by e Z X (3.16) F (B) := 1B (xfk )Fk (dx), B ∈ B(Re ) k=1

R

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

13

where F1 , . . . , Fe are measures on (R, B(R)) satisfying (3.17) (3.18)

Fk ({0}) = 0, k = 1, . . . , e Z (|x|2 ∧ 1)Fk (dx) < ∞, k = 1, . . . , e R

and where the (fk )k=1,...,e denote the unit vectors in Re . Note that definition (3.16) implies Z g(x)F (dx) =

(3.19) Re

e Z X k=1

g(xfk )Fk (dx)

R

e for any nonnegative Semeasurable function g : R → R, in particular, the esupport of F is contained in k=1 span{fk }, the union of the coordinate axes in R . For each k = 1, . . . , e let δk : Hβ → Hβ00 be a map. We define γ : Hβ × Re → Hβ00 as

γ(h, x) :=

(3.20)

e X

δk (h)xk 1span{fk } (x).

k=1

Then, equation (1.9) corresponds to the situation where the term structure model is driven by several real-valued, independent L´evy processes. For all h ∈ Hβ and Rξ ξ ∈ R+ we set ∆k (h)(ξ) := − 0 δk (h)(η)dη, k = 1, . . . , e. 3.4. Assumption. We assume there exist constants N,  > 0 such that for all k = 1, . . . , e we have Z (3.21) ezx Fk (dx) < ∞, z ∈ [−(1 + )N, (1 + )N ] {|x|>1}

(3.22)

|∆k (h)(ξ)| ≤ N,

h ∈ Hβ , ξ ∈ R+

a constant L > 0 such that (3.2) and (3.23)

kδk (h1 ) − δk (h2 )kβ 0 ≤ Lkh1 − h2 kβ ,

k = 1, . . . , e

are satisfied for all h1 , h2 ∈ Hβ , and a constant M > 0 such that (3.4) and (3.24)

kδk (h)kβ 0 ≤ M,

k = 1, . . . , e

are satisfied for all h ∈ Hβ . Now, we obtain the statement of [19, Thm. 4.6] as a corollary. 3.5. Corollary. Suppose Assumption 3.4 is fulfilled. Then, for each initial curve h0 ∈ L2 (Ω, F0 , P; Hβ ) there exists a unique adapted, c` adl` ag, mean square continuous Hβ -valued solution (ft )t≥0 for the HJM equation (1.10) with f0 = `h0 satisfying (3.13), and there exists a unique adapted, c` adl` ag, mean square continuous mild and weak Hβ -valued solution (rt )t≥0 for the HJMM equation (1.9) with r0 = h0 satisfying (3.14), which is given by rt := πUt ft , t ≥ 0. Moreover, the implied bond market (1.4) is free of arbitrage. Proof. Using (3.22), the measurable function Φ : Re → R+ defined as Φ(x) := N

e X k=1

|xk |1span{fk } (x),

x ∈ Re

14

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

satisfies (3.1). For each k = 1, . . . , e and every m ∈ N with m ≥ 2 we have, by (3.18) and (3.21), (3.25) Z Z Z  m zx m |zx| |x e |Fk (dx) ≤ |x| e Fk (dx) ≤ |x|m e(1+ 2 )N |x| Fk (dx) R R Z R Z m! m |x| Fk (dx) +  m e(1+)N |x| Fk (dx) < ∞ ≤2 (2N) {|x|> ln2 } {|x|≤ ln2 } (1+

2

)N

(1+

2

)N

 2 )N ].

 2 )N, (1

+ Taking into account (3.19), (3.25), relations for all z ∈ [−(1 + (3.23), (3.24) imply (3.3), (3.5). Furthermore, (3.25) and Lebesgue’s theorem show that the cumulant generating functions Z Ψk (z) := (ezx − 1 − zx)Fk (dx), k = 1, . . . , e R

belong to class C ∞ on the open interval (−(1 + 4 )N, (1 + 4 )N ) with derivatives Z 0 Ψk (z) = x(ezx − 1)Fk (dx), R Z (m) Ψk (z) = xm ezx Fk (dx), m ≥ 2. R

Therefore, and because of (3.19), we can, for an arbitrary h ∈ Hβ , write α2 (h), which is defined in (3.6), as  Z •  e X α2 (h) = − δk (h)Ψ0k − δk (η)dη . k=1

0

Hence, α2 (h) is absolutely continuous with weak derivative (3.7). Consequently, Assumption 3.1 is fulfilled and Theorem 3.3 applies.  Note that the boundedness assumptions (3.4), (3.5) of Theorem 3.3 resp. (3.4), (3.24) of Corollary 3.5 cannot be weakened substantially. For example, for arbitrage free term structure models driven by a single Brownian motion, it was shown in [31, Sec. 4.7] that for the simple case of proportional volatility, that is σ(h) = σ0 h for some constant σ0 > 0, solutions necessarily explode. We mention, however, that [36, Sec. 6] contains some existence results for L´evy term structure models with linear volatility. 4. Positivity preserving term structure models driven by Wiener process and Poisson measures In applications, we are often interested in term structure models producing positive forward curves. In this section, we characterize HJMM forward curve evolutions of the type (1.9), which preserve positivity, by means of the characteristic coefficients of the SPDE. In the case of short rate models this can be characterized by the positivity of the short rate, a one-dimensional Markov process. In case of a infinitefactor evolution, as described by a generic HJMM equation (see for instance [2]), this problem is much more delicate. Indeed, one has to find conditions such that a Markov process defined by the HJMM equation (on a Hilbert space of forward rate curves) stays in a “small” set of curves, namely the convex cone of positive curves bounded by a non-smooth set. Our strategy to solve this problem is the following: first we show by general semimartingale methods necessary conditions for positivity. These necessary conditions are basically the described by the facts that the Itˆ o drift is inward-pointing and that the volatilities are parallel at the boundary of the set of non-negative functions. Taking those conditions we can also prove that the Stratonovich drift is inward pointing, since parallel volatilities produce parallel

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

15

Stratonovich corrections (a fact which is not true for general closed convex sets but true for the set of non-negative functions P ). Then we reduce the sufficiency proof to two steps: first we essentially apply results from [34] in order to solve the pure diffusion case and then we slowly switch on the jumps to see the general result. Let Hβ be the space of forward curves introduced in Section 2 for some fixed β > 0. We introduce the half spaces Hξ+ := {h ∈ Hβ | h(ξ) ≥ 0},

ξ ∈ R+

and define the closed, convex cone P :=

\

Hξ+

ξ∈R+

consisting of all nonnegative forward curves from Hβ . In what follows, we shall use that, by the continuity of the functions from Hβ , we can write P as \ P := Hξ+ . ξ∈(0,∞)

Furthermore the edges ∂Pξ := {h ∈ P | h(ξ) = 0},

ξ ∈ (0, ∞).

First, we consider the positivity problem for general forward curve evolutions, where the HJM drift condition (1.5) is not necessarily satisfied, and afterwards we apply our results to the arbitrage free situation. We emphasize that, in the sequel, we assume the existence of solutions. Sufficient conditions for existence and uniqueness are provided in Appendix A for general stochastic partial differential equations and in the previous Section 3 for the HJMM term structure equation (1.9). Again, the framework is the same as in Appendix A with H = Hβ being the space of forward curves, equipped with the strongly continuous semigroup (St )t≥0 d . of shifts, which has the infinitesimal generator A = dξ At first glance, it looks reasonable to treat the positivity problem by working with weak solutions on Hβ . However, this is unfeasible, because – as the next lemma reveals – the point evaluations at ξ ∈ (0, ∞), i.e., a linear functional ζ ∈ Hβ such d ∗ ) ) of the that h(ξ) = hζ, hi for all h ∈ Hβ , do never belong to the domain D(( dξ adjoint operator. d 4.1. Lemma. For each ξ ∈ (0, ∞) the linear functional h 7→ h0 (ξ) : D( dξ ) → R is unbounded.

Proof. Let ξ ∈ (0, ∞) be arbitrary. We define ψ : R → R as ( 1 e− η , η > 0 ψ(η) := 0, η ≤ 0. Furthermore, we define ϕ : R → R as ϕ(η) := eψ(1 − η 2 ), and for each n ∈ N we define the mollifier ϕn : R → R by ϕn (η) := nϕ(nη). Then, for each n ∈ N we have ϕn ∈ C ∞ (R) with supp(ϕn ) ⊂ [− n1 , n1 ], see, e.g., [41, p. 81,82]. There exists n0 ∈ N such thatR ξ − n1 > 0 for all n ≥ n0 . For each n ≥ n0 we define gn : R → R as η gn (η) := 0 ϕn (ζ − ξ)dζ and hn : R+ → R as hn := gn |R+ . d Then we have hn ∈ D( dξ ) with |h0n (η)| ≤ n, η ∈ [ξ − n1 , ξ + n1 ] and |h0n (ξ)| = n for all n ≥ n0 . The estimate Z ξ+ n1 2 2 khn kβ = |h0n (η)|2 eβη dη ≤ n2 eβ(ξ+1) = 2eβ(ξ+1) n, n ≥ n0 1 n ξ− n d shows that the linear functional h 7→ h0 (ξ) : D( dξ ) → R is unbounded.



´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

16

Therefore treating the positivity problem with weak solutions does not bring an immediate advantage, hence we shall work with mild solutions on Hβ . Let α : Hβ → Hβ , σ : Hβ → L02 (Hβp ) and γ : Hβ × E → Hβ be given. For each j we define σ j : Hβ → Hβ as σ j (h) := λj σ(h)ej . We assume that for each h0 ∈ P the HJM equation  P j j   dft = U−t R `α(πUt ft )dt + j U−t `σ (πUt ft )dβt + E U−t `γ(πUt ft− , x)(µ(dt, dx) − F (dx)dt) (4.1)   f0 = `h0 , has at least one solution (ft )t≥0 . Then, because of (2.7), the transformation rt := πUt ft , t ≥ 0 is a mild solution of the HJMM equation (4.2) ( drt r0

=

d rt + α(rt ))dt + ( dξ

P

j

σ j (rt )dβtj +

R E

γ(rt− , x)(µ(dt, dx) − F (dx)dt)

= h0 .

4.2. Definition. The HJMM equation (4.2) is said to be positivity preserving if for all h0 ∈ L2 (Ω, F0 , P; HT β ) with P(h0 ∈ P ) = 1 and every solution (ft )t≥0 of (4.1) with f0 = `h0 we have P( t∈R+ {rt ∈ P }) = 1, where rt := πUt ft , t ≥ 0. 4.3. Remark. Note that seemingly weaker condition that P({rt ∈ P }) = 1 for all t ∈ R+ is equivalent to condition of the previous definition due to the c` adl` ag property of the trajectories. 4.4. Definition. The HJMM equation (4.2) is said to be locally positivity preserving if for all h0 ∈ L2 (Ω, F0 , P; Hβ ) with P(h0 ∈ P ) = 1 and every solution (ft )t≥0 of T (4.1) with f0 = `h0 there exists a strictly positive stopping time τ such that P( t∈R+ {rt∧τ ∈ P }) = 1, where rt := πUt ft , t ≥ 0. 4.5. Lemma. Let h0 ∈ P be arbitrary and let (ft )t≥0 be a solution for (4.1) with f0 = `h0 . Set rt := πUt ft , t ≥ 0. The following two statements are equivalent: T (1) We have P( t∈R+ {rt ∈ P }) = 1. T (2) We have P( t∈[0,T ] {ft (T ) ≥ 0}) = 1 for all T ∈ (0, ∞). Proof. The claim follows, because the processes (rt )t≥0 and (ft (T ))t∈[0,T ] for an arbitrary T ∈ (0, ∞) are c` adl` ag, and because the functions from Hβ are continuous.  4.6. Assumption. We assumeR that the maps α : Hβ → Hβ and σ : Hβ → L02 (Hβ ) are continuous and that h 7→ B γ(h, x)F (dx) is continuous on Hβ for all B ∈ E with F (B) < ∞. 4.7. Proposition. Suppose Assumption 4.6 is fulfilled. If equation (4.2) is positivity preserving, then we have Z (4.3) γ(h, x)(ξ)F (dx) < ∞, for all ξ ∈ (0, ∞), h ∈ ∂Pξ E Z (4.4) α(h)(ξ) − γ(h, x)(ξ)F (dx) ≥ 0, for all ξ ∈ (0, ∞), h ∈ ∂Pξ E

(4.5)

σ j (h)(ξ) = 0, for all ξ ∈ (0, ∞), h ∈ ∂Pξ and all j

(4.6)

h + γ(h, x) ∈ P, for all h ∈ P and F -almost all x ∈ E.

4.8. Remark. Notice that, by H¨ older’s inequality, Assumption 4.6 is implied by Assumptions A.5, A.6, and therefore in particular by Assumption 3.1.

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

17

4.9. Remark. In view of (4.3), observe that condition (4.6) implies γ(h, x)(ξ) ≥ 0, for all ξ ∈ (0, ∞), h ∈ ∂Pξ and F -almost all x ∈ E.

(4.7)

4.10. Remark. Notice that conditions (4.3) and (4.4) can be unified to Z |γ(h, x)(ξ)|F (dx) ≤ α(h)(ξ) E

for all ξ ≥ 0 and h ∈ ∂Pξ . Proof. Let h0 ∈ P be arbitrary and let (ft )t≥0 be a solution for (4.1) with f0 = `h0 . By Lemma 4.5, for each T ∈ (0, ∞) and every stopping time τ ≤ T we have P(fτ (T ) ≥ 0) = 1.

(4.8)

Let φ ∈ U00 be a linear functional such that φj := φej 6= 0 for only finitely many j, and let ψ : E → R be a measurable function of the form ψ = c1B with c > −1 and B ∈ E satisfying F (B) < ∞. Let Z be the Dol´eans-Dade Exponential  X Z •Z j j φ β + ψ(x)(µ(ds, dx) − F (dx)ds) , t ≥ 0. Zt = E 0

j

E

t

By [25, Thm. I.4.61] the process Z is a solution of Z tZ X Z t j j Zt = 1 + φ Zs dβs + Zs− ψ(x)(µ(ds, dx) − F (dx)ds), 0

j

0

t≥0

E

and, since ψ > −1, the process Z is a strictly positive local martingale. There exists a strictly positive stopping time τ1 such that Z τ1 is a martingale. Due to the method of the moving frame, see [20], we can use standard stochastic analysis, to proceed further. For an arbitrary T ∈ (0, ∞), integration by parts yields (see [25, Thm. I.4.52]) Z t Z t ft (T )Zt = fs− (T )dZs + Zs− dfs (T ) + hf (T )c , Z c it 0 0 (4.9) X + ∆fs (T )∆Zs , t ≥ 0. s≤t

Taking into account the dynamics Z t XZ t ft (T ) = `h0 (T ) + U−s `α(πUs fs )(T )ds + U−s `σ j (πUs fs )(T )dβsj 0

(4.10)

j

0

Z tZ U−s `γ(πUs fs− , x)(T )(µ(ds, dx) − F (dx)ds),

+ 0

t≥0

E

we have hf (T )c , Z c it =

(4.11)

X

φj

(4.12)

t

Zs U−s `σ j (πUs fs )(T )ds,

t≥0

0

j

X

Z

Z tZ ∆fs (T )∆Zs =

Zs− ψ(x)U−s `γ(πUs fs− , x)(T )µ(ds, dx), 0

s≤t

t ≥ 0.

E

Incorporating (4.10), (4.11) and (4.12) into (4.9), we obtain  Z t X ft (T )Zt = Mt + Zs− U−s `α(πUs fs− )(T ) + φj U−s `σ j (πUs fs− )(T ) 0

(4.13) Z + E

j

 ψ(x)U−s `γ(πUs fs− , x)(T )F (dx) ds,

t≥0

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

18

where M is a local martingale with M0 = 0. There exists a strictly positive stopping time τ2 such that M τ2 is a martingale. By Assumption 4.6 there exist strictly positive stopping times τ3 , τ4 , τ5 and constants α ˜, σ ˜ (φ), γ˜ (ψ) > 0 such that |U−(t∧τ3 ) `α(πUt∧τ3 f(t∧τ3 )− )(T )| ≤ α ˜, t ≥ 0 X j j ≤σ φ U `σ (πU f )(T ) t∧τ4 (t∧τ4 )− −(t∧τ4 ) ˜ (φ), t ≥ 0 j Z ψ(x)U−(t∧τ5 ) `γ(πUt∧τ5 f(t∧τ5 )− , x)(T )F (dx) ≤ γ˜ (ψ),

t ≥ 0.

E

Let B := {x ∈ E : h0 + γ(h0 , x) ∈ / P }. In order to prove (4.6), it suffices, since F is σ-finite, to show that F (B ∩ C) = 0 for all C ∈ E with F (C) < ∞. Suppose, on the contrary, there exists C ∈ E with F (C) < ∞ such that F (B ∩ C) > 0. By the continuity of the functions from Hβ , there exists T ∈ (0, ∞) such that F (BT ∩ C) > 0, where BT := {x ∈ E : h0 (T ) + γ(h0 , x)(T ) < 0}. We obtain Z Z γ(h0 , x)(T )F (dx) ≤ (h0 (T ) + γ(h0 , x)(T ))F (dx) < 0. BT ∩C

BT ∩C

By Assumption 4.6 and left continuity of the process f.− , there exist η > 0 and a strictly positive stopping time τ6 ≤ T such that Z U−(t∧τ6 ) `γ(πU(t∧τ6 ) f(t∧τ6 )− , x)(T )F (dx) ≤ −η, t ≥ 0. BT ∩C

V6 ˜ Let φ := 0, ψ := α+1 i=1 τi . Taking expectation in (4.13) we η 1BT ∩C and τ := obtain E[fτ (T )Zτ ] < 0, implying P(fτ (T ) < 0) > 0, which contradicts (4.8). This yields (4.6). From now on, we assume that h0 ∈ ∂PT for an arbitrary T ∈ (0, ∞). Suppose that σ j (h0 )(T ) 6= 0 for some j. By the continuity of σ (see Assumption 4.6) there exist η > 0 and a strictly positive stopping time τ6 ≤ T such that |U−(t∧τ6 ) `σ j (πUt∧τ6 f(t∧τ6 )− )(T )| ≥ η,

t ≥ 0.

˜ Let φ ∈ U00 be the linear functional with φj = −sign(σ j (h0 )(T )) α+1 and φk = 0 η V6 for k 6= j. Furthermore, let ψ := 0 and τ := i=1 τi . Taking expectation in (4.13) yields E[fτ (T )Zτ ] < 0, implying P(fτ (T ) < 0) > 0, which contradicts (4.8). This proves (4.5). R Now suppose E γ(h0 , x)(T )F (dx) = ∞. Using Assumption 4.6, relation (4.7) and the σ-finiteness of F , there exist B ∈ E with F (B) < ∞ and a strictly positive stopping time τ6 ≤ T such that Z 1 − U−(t∧τ6 ) `γ(πUt∧τ6 f(t∧τ6 )− , x)(T )F (dx) ≤ −(˜ α + 1), t ≥ 0. 2 B V6 Let φ := 0, ψ := − 21 1B and τ := i=1 τi . Taking expectation in (4.13) we obtain E[fτ (T )Zτ ] < 0, implying P(fτ (T ) < 0) > 0, which contradicts (4.8). This yields (4.3). Since F is σ-finite, there exists a sequence (Bn )n∈N ⊂ E with Bn ↑ E and F (Bn ) < ∞, n ∈ N. Next, we show for all n ∈ N the relation Z (4.14) α(h0 )(T ) + ψn (x)γ(h0 , x)(T )F (dx) ≥ 0, E

where ψn := −(1 − 1 Suppose, on the contrary, that (4.14) is not satisfied for some n ∈ N. Using Assumption 4.6, there exist η > 0 and a strictly positive 1 n ) Bn .

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

19

stopping time τ6 ≤ T such that U−(t∧τ6 ) `α(πUt f(t∧τ6 )− )(T ) Z + ψn (x)U−(t∧τ6 ) `γ(πUt∧τ6 f(t∧τ6 )− , x)(T )F (dx) ≤ −η,

t ≥ 0.

E

V6 Let φ := 0 and τ := i=1 τi . Taking expectation in (4.13) we obtain E[fτ (T )Zτ ] < 0, implying P(fτ (T ) < 0) > 0, which contradicts (4.8). This yields (4.14). By (4.14), (4.3) and Lebesgue’s theorem, we conclude (4.4).  We shall now present sufficient conditions for positivity preserving term structure models. In the sequel, we suppose that Assumptions A.5, A.6 are fulfilled, which ensures existence and uniqueness of solutions by Theorem A.7. 4.11. Lemma. Suppose Assumptions A.5, A.6 are fulfilled. If equation (4.2) is locally positivity preserving and we have (4.6), then equation (4.2) is positivity preserving. Proof. Let h0 ∈ L2 (Ω, F0 , P; Hβ ) be arbitrary. Moreover, let (rt )t≥0 be the mild solution for (4.2) with r0 = h0 . We define the stopping time τ0 := inf{t > 0 : rt ∈ / P }.

(4.15)

By the closedness of P and (4.6) we have rτ0 ∈ P on {τ0 < ∞}. We claim that τ0 = ∞. Assume, on the contrary, that (4.16)

P(τ0 < N ) > 0

for some N ∈ N. Let τ1 be the bounded stopping time τ1 := τ0 ∧ N . We define the ˜ t )t≥0 , the new Q-Wiener process W ˜ and the new Poisson random new filtration (F 2 ˜ 0 , P; Hβ ), because, by (3.14), measure µ ˜ as in Lemma A.9. Note that rτ1 ∈ L (Ω, F we have   E[krτ1 k2β ] ≤ E sup krt k2β < ∞. t∈[0,N ]

˜ t )-adapted process r˜t := rτ +t is the unique mild solution By Lemma A.9, the (F 1 for ( R P d r˜t + α(˜ rt ))dt + j σ j (˜ rt )dβ˜tj + E γ(˜ rt− , x)(˜ µ(dt, dx) − F (dx)dt) d˜ rt = ( dξ r˜0

= rτ1 .

Since equation (4.2) is locally positivity preserving and T P(rτ1 ∈ P ) = 1, there exists a strictly positive stopping time τ2 such that P( t∈R+ r˜t∧τ2 ∈ P ) = 1. Since {τ0 < N } ⊂ {τ0 = τ1 }, we obtain rτ0 +t ∈ P

on [0, τ2 ] ∩ {τ0 < N },

which is a contradiction because of (4.16) and the definition (4.15) of τ0 . Consequently, we have τ0 = ∞.  4.12. Assumption. We assume σ ∈ C 2 (H; L02 (H)), and that the vector fields 1X Dσ j (h)σ j (h), h 7→ σ j (h) h 7→ σ 0 (h) := α(h) − 2 j are globally Lipschitz from H to H. 4.13. Lemma. Suppose Assumption 4.12 and relation (4.5) are fulfilled. Then we have X
Dσ j (h)σ j (h) (ξ) = 0, for all ξ ∈ (0, ∞), h ∈ ∂Pξ . j

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

20


Proof. It suffices to show Dσ j (h)σ j (h) (ξ) = 0 for all h ∈ ∂Pξ and all j. Therefore let j be fixed and denote σ = σ j . By assumption for all h ≥ 0 with h(ξ) = 0 we have that σ(h)(ξ) = 0. In other words the volatility vector field σ is parallel to the boundary at boundary elements of P . We denote the local flow of the Lipschitz vector field σ by Fl being defined on a small time interval ] − , [ around time 0 and a small neighborhood of each element h ∈ P . We state first that the flow Fl leaves the set P invariant, i.e., Flt (h) ≥ 0 if h ≥ 0, by convexity and closedness of the cone of positive functions due to [40]. Indeed, P is a closed and convex cone, whose supporting hyperplanes l (i.e., a linear functional l is called supporting hyperplane of P at h if l(P ) ≥ 0 and l(h) = 0) are given by appropriate positive measures µ on R+ via Z l(h) = h(ξ)µ(dξ), R+

whence condition (4) from [40] is fulfilled due to (4.5). Next we show that even more holds: the solution Flt (h) evaluated at ξ vanishes if h(ξ) = 0, which we show directly. Indeed, let us additionally fix h ∈ ∂Pξ , i.e., h ≥ 0 and h(ξ) = 0. Looking now at the Picard-Lindel¨ of approximation scheme Z t c(n+1) (t) = h + σ(c(n) (s))ds 0 (n)

(0)

with c (0) = h and c (s) = h for s, t ∈] − , [ and n ≥ 0, we see by induction that under our assumptions c(n) (t)(ξ) = 0 for all n ≥ 0 and t ∈] − , [ for the given fixed element h. Consequently – as n → ∞ – we obtain that Flt (h)(ξ) = 0, which is the limit of c(n) (t). Therefore (Dσ(h)σ(h))(ξ) =

d |s=0 σ(Fls (h))(ξ) = 0, ds

since Flt (h) ≥ 0 by invariance and Flt (h)(ξ) = 0 by the previous consideration lead to σ(Flt (h))(ξ) = 0 for t ∈] − , [. Notice that we did not need the global Lipschitz property of the Stratonovich correction for the proof of this lemma.  Before we show sufficiency for the HJMM equation (4.2) with jumps, we consider the pure diffusion case. Notice that due to Lemma 4.13 the condition (4.4) is in fact equivalent to the very same condition formulated with the Stratonovich drift σ 0 instead of α, since the Stratonovich correction vanishes at the boundary of P . In order to treat the pure diffusion case, we apply [34], which, by using the support theorem provided in [33], offers a general characterization of stochastic invariance of closed sets for SPDEs. Other results for positivity preserving SPDEs, where, in contrast to our framework, the state space is an L2 -space, can be found in [27] and [30]. The results from [30] have been used in [36] in order to derive some positivity results for L´evy term structure models on L2 -spaces. 4.14. Proposition. Suppose Assumptions A.6, 4.12 are fulfilled and γ ≡ 0. If conditions (4.4), (4.5) are satisfied, then equation (4.2) is positivity preserving. Proof. First we assume that the vector fields σ j for j ≥ 0 are bounded in order to apply Nakayama’s beautiful support theorem from [34]. Namely, for n ∈ N we choose a function ψn ∈ C ∞ (H; [0, 1]) such that ψn ≡ 1 on Bn (0) and supp(ψn ) ⊂ Bn+1 (0)

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

21

and define the vector fields αn (h) := ψn (h)α(h), σnj (h) := ψn (h)σ j (h),

j≥1 1X σn0 (h) := ψn (h)α(h) − Dσnj (h)σnj (h), 2 j

which again satisfy Assumptions A.6, 4.12 as well as conditions (4.4), (4.5). We therefore show that the semigroup Nagumo’s condition (3) from [33, Prop. 1.1] is fulfilled due to conditions (4.4) and (4.5). Introducing the distance dP (h) from P as minimal distance of h from P , we can formulate Nagumo’s condition (3) as 1 lim inf dP (St h + tσ 0 (h) + tσ(h)u) = 0 t↓0 t for all u ∈ U0 and h ∈ P . Fix now h ∈ P and u ∈ U0 and introduce the abbreviation σ = σ 0 + σ(·)u, then obviously

Flσt (h) − h σ 0

,

kSt h + tσ (h) + tσ(h)u − St Flt (h)kβ = t σ(h) − St

t β which means that 1 lim kSt h + tσ 0 (h) + tσ(h)u − St Flσt (h)kβ = 0. t↓0 t Hence Nagumo’s condition can be equivalently formulated as 1 lim inf dP (St Flσt (h)) = 0, t↓0 t for the particular choice of u and h ∈ P . Due to conditions (4.4) and (4.5) the semiflow Flσ leaves P invariant by [40], the semigroup St certainly, too, therefore dP (St Flσt (h)) = 0 and whence Nagumo’s condition is more than satisfied.  4.15. Remark. Having in mind the method of the moving frame, we could also have shown the Nagumo condition by argueing with time-dependent versions of Section 4 of [38] or [40]. The method of the moving frame would work well in this particular case, since the convex set ∩t∈R St P of functions positive on the whole real line R is invariant under the action of the shift S. 4.16. Proposition. Suppose Assumptions A.5, A.6, 4.12 and conditions (4.3), (4.4), (4.5), (4.6) are fulfilled. Then, equation (4.2) is positivity preserving. Proof. Since the measure F is σ-finite, there exists a sequence (Bn )n∈N ⊂ E with Bn ↑ E and F (Bn ) < ∞ for all n ∈ N. Let h0 ∈ L2 (Ω, F0 , P; Hβ ) be arbitrary. Relations (4.4), (4.7), (4.5), Proposition 4.14 and (4.6) together with the closedness of P yield that, for each n ∈ N, the mild solution (rtn )t≥0 of the stochastic partial differential equation R  n P j d  drt = ( dξRrtn + α(rtn ) − Bn γ(rtn , x))dt + j σ j (rtn )dβt n (4.17) + Bn γ(rt− , x)µ(dt, dx)  n r0 = h0 T n satisfies P( t∈R+ rt∧τ ∈ P ) = 1, where τ denotes the strictly positive stopping time τ := inf{t > 0 : µ([0, t] × Bn ) = 1}.

22

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

By virtue of Lemma 4.11, for each n ∈ N equation (4.17) is positivity preserving. According to [20, Prop. 9.1] we have   E sup krt − rtn k2β → 0 for all T ∈ R+ , t∈[0,T ]

proving that equation (4.2) is positivity preserving.



Finally the next theorem states the sufficient conditions under which we can conclude that the solution of equation (4.2) is positivity preserving. 4.17. Theorem. Suppose Assumptions A.5, A.6, 4.12 are fulfilled. Then, for each initial curve h0 ∈ L2 (Ω, F0 , P; Hβ ) there exists a unique adapted, c` adl` ag, mean square continuous Hβ -valued solution (ft )t≥0 for the HJM equation (4.1) with f0 = `h0 satisfying (3.13), and there exists a unique adapted, c` adl` ag, mean square continuous mild and weak Hβ -valued solution (rt )t≥0 for the HJMM equation (4.2) with r0 = h0 satisfying (3.14), which is given by rt := πUt ft , t ≥ 0. Moreover, equation (4.2) is positivity preserving if and only if we have (4.3), (4.4), (4.5), (4.6). Proof. The statement follows from Theorem A.7, Proposition 4.7 (see also Remark 4.8) and Proposition 4.16.  4.18. Remark. Note that Theorem 4.17 is also valid on other state spaces. The only essential requirement is that the Hilbert space H consists of continuous, real-valued functions on which the point evaluations are continuous functionals, and that the shift semigroup extends to a strongly continuous group in the sense of Assumption A.4. 4.19. Remark. For the particular situation where equation (4.2) has no jumps, Theorem 4.17 corresponds to the statement of [30, Thm. 3], where positivity on weighted L2 -spaces is investigated. Since point evaluations are discontinuous functionals on L2 -spaces, the conditions in [30] are formulated by taking other appropriate functionals. We shall now consider the arbitrage free situation. Let α = αHJM : Hβ → Hβ in (4.2) be defined according to the HJM drift condition (1.5). 4.20. Proposition. Conditions (4.3), (4.4), (4.5), (4.6) are satisfied if and only if we have (4.5), (4.6) and (4.18)

γ(h, x)(ξ) = 0,

ξ ∈ (0, ∞), h ∈ ∂Pξ and F -almost all x ∈ E.

Proof. Provided (4.5), (4.6) are fulfilled, conditions (4.3), (4.4) are satisfied if and only if we have (4.3) and Z (4.19) − γ(h, x)(ξ)eΓ(h,x)(ξ) F (dx) ≥ 0, ξ ∈ (0, ∞), h ∈ ∂Pξ E

because the drift α is given by (1.5). By (4.7), relations (4.3), (4.19) are fulfilled if and only if we have (4.18).  Now let, as in Section 3, coefficients σ : Hβ → L02 (Hβ0 ) and γ : Hβ × E → Hβ00 be given, where β 0 > β is a real number. 4.21. Theorem. Suppose Assumption 3.1 is fulfilled, suppose furthermore that σ ∈ C 2 (H; L02 (H)), and that the vector field 1X Dσ j (h)σ j (h) h 7→ − 2 j

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

23

is globally Lipschitz from H to H. Then, the statement of Theorem 3.3 is valid, and, in addition, the HJMM equation (1.9) is positivity preserving if and only if we have (4.5), (4.6), (4.18). Proof. The statement follows from Theorem 3.3, Theorem 4.17 and Proposition 4.20.  Finally, let us consider the L´evy case, treated at the end of Section 3. In this framework, the following statement is valid. 4.22. Proposition. Conditions (4.5), (4.6) and (4.18) are satisfied if and only if we have (4.5) and (4.20) h + δk (h)x ∈ P, (4.21) δk (h)(ξ) = 0,

h ∈ P, k = 1, . . . , e and Fk -almost all x ∈ R ξ ∈ (0, ∞), h ∈ ∂Pξ and all k = 1, . . . , e with Fk (R) > 0.

Proof. The claim follows from the definition (3.16) of F and the definition (3.20) of γ.  4.23. Corollary. Suppose Assumption 3.4 is fulfilled, suppose furthermore that σ ∈ C 2 (H; L02 (H)), and that the vector field 1X Dσ j (h)σ j (h), h 7→ − 2 j is globally Lipschitz from H to H. Then, the statement of Corollary 3.5 is valid, and, in addition, the HJMM equation (1.9) is positivity preserving if and only if we have (4.5), (4.20), (4.21). Proof. The assertion follows from Theorem 4.21 and Proposition 4.22.



Our above results on arbitrage free, positivity preserving term structure models apply in particular for local state dependent volatilities. The following two results are obvious. 4.24. Proposition. Suppose for all j there are σ ˜ j : R+ × R → R, and γ˜ : R+ × R × E → R such that (4.22)

σ j (h)(ξ) = σ ˜ j (ξ, h(ξ)),

(4.23)

γ(h, x)(ξ) = γ˜ (ξ, h(ξ), x),

(h, ξ) ∈ Hβ × R+ ,

for all j

(h, x, ξ) ∈ Hβ × E × R+ .

Then, conditions (4.5), (4.6), (4.18) are fulfilled if and only if (4.24)

σ ˜ j (ξ, 0) = 0,

(4.25)

y + γ˜ (ξ, y, x) ≥ 0,

(4.26)

γ˜ (ξ, 0, x) = 0,

ξ ∈ (0, ∞),

for all j

ξ ∈ (0, ∞), y ∈ R+ and F -almost all x ∈ E

ξ ∈ (0, ∞) and F -almost all x ∈ E.

L´evy term structure models with local state dependent volatilities have been studied in [36] and [28]. In the framework of Proposition 4.22 we obtain: 4.25. Proposition. Suppose for all j there are σ ˜ j : R+ × R → R, and for all ˜ k = 1, . . . , e there are δk : R+ × R → R such that we have (4.22) and (4.27) δk (h)(ξ) = δ˜k (ξ, h(ξ)), (h, ξ) ∈ Hβ × R+ , k = 1, . . . , e. Then, conditions (4.5), (4.20), (4.21) are fulfilled if and only if we have (4.24) and (4.28) y + δ˜k (ξ, y)x ≥ 0, (4.29) δ˜k (ξ, 0) = 0,

ξ ∈ (0, ∞), y ∈ R+ , k = 1, . . . , e and Fk -almost all x ∈ R

ξ ∈ (0, ∞) and all k = 1, . . . , e with Fk (R) > 0.

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

24

Section 5 in [36] contains some positivity results for L´evy driven term structure models on weighted L2 -spaces. Using Proposition 4.25, we can derive the analogous statements of [36, Thm. 4] on our Hβ -spaces. 5. The Brody-Hughston equation: Existence and uniqueness Let (H, k · k) now denote a state (Hilbert) space of continuous, integrable realvalued maps on R, where the shift semigroup acts as a strongly continuous group, for instance H 1 (R). We need one notation for the sake of simplicity: Let ρ denote a probability density on R+ , extended by 0 to the whole real line, and η ∈ L1 (R, ρ(·)du). R 5.1. Assumption. We assume E kγ(0, x)k2 F (dx) < ∞ and that there exists a constant L > 0 such that (5.1) Z (5.2)

kσ(h1 ) − σ(h2 )kL02 (H) ≤ Lkh1 − h2 k,  21 ≤ Lkh1 − h2 k kγ(h1 , x) − γ(h2 , x)k2 F (dx)

E

for all h1 , h2 ∈ H. Furthermore we assume for all ρ ∈ H that Z ∞ (5.3) σ(ρ)(u)du = 0, 0

and we assume that (5.4)

σ(ρ)(ξ) = 0

for all ρ ∈ P , ξ ≥ 0 and ρ(ξ) = 0. For the jump fields we assume that (5.5) (5.6)

ρ + γ(ρ, x) ∈ P, for all ρ ∈ P and F -almost all x ∈ E, Z − γ(ρ, x)F (dx)(ξ) ≥ 0, for all ξ ∈ (0, ∞), ρ ∈ ∂Pξ , E

and finally that Z

∞

(5.7)

γ(ρ, x)(u)du = 0 0

for all ρ ∈ H and F -almost all x ∈ E. Under theses assumptions we can prove the following theorem: 5.2. Theorem. The following equation, which we call henceforward Brody-Hughston equation,  P j j d   dρt = ( dξ ρt + ρt (0)ρt )dt + j σ (ρt )dβt +γ(ρt− , x)(µ(dt, dx) − F (dx)dt) (5.8)   ρ0 ∈ H, has a unique adapted, c` adl` ag, mean square continuous mild and weak solution for all times in H, which leaves the set of densities invariant. Furthermore Z ∞ P (t, T ) = ρ(t, u)du T −t

for 0 ≤ t ≤ T defines an arbitrage-free evolution of bond prices, i.e. the discounted bond price processes  Z t Z ∞ exp − ρs (0)ds ρ(t, u)du 0

are local martingales for 0 ≤ t ≤ T .

T −t

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

25

Proof. The proof is a direct application of Theorem 4.17 (see also Remark 4.18). The mass of ρt is preserved since applying the linear functional Z ∞ ρ 7→ ρ(u)du 0

makes the right hand side vanish.



5.3. Remark. Notice the conceptual simplicity of the Brody-Hughston equation (5.8) in contrast to the HJM equation. In particular, adding jumps to the BrodyHughston equation is less delicate than in the HJMM case. Remark also that positivity is the crucial issue for the HJMM equation as well as for the Brody-Hughston equation. 5.4. Remark. We can define the average of η at ρ Z ∞ η= η(u)ρ(u)du. 0

We suppress in this notation the dependence on ρ, but it should be clear at every moment, where it appears, which ρ is meant. Vector fields, volatilities or jump fields, satisfying Assumptions 5.1 can then be chosen of the form ρ 7→ (a(ρ) − (a(ρ))ρ, for some a : H → H appropriately chosen. In this case, i.e., σ j and γ chosen of the previous type, one can try to divide the equation by ρ in order to show positivity directly, which works up to some regularity questions. Our approach chosen here is more general since we do not need to assume that the vector fields factor by ρ. Appendix A. Stochastic partial differential equations driven by Wiener process and Poisson measures For convenience of the reader, we provide the crucial results on stochastic partial differential equations driven by Wiener process and Poisson measures in this appendix. For this purpose, we follow [20], where we understand stochastic partial differential equations – in this paper, the HJMM equation (1.9) – as time-dependent transformations of stochastic differential equations – in this text, the HJM equation (1.10). Other references for existence and uniqueness results on stochastic partial differential equations driven by Wiener process and Poisson measures are [1] and [29]. Let H denote a separable Hilbert space with inner product h·, ·i and associated norm k · k. Furthermore, let (St )t≥0 be a C0 -semigroup on H with infinitesimal generator A : D(A) ⊂ H → H. We denote by A∗ : D(A∗ ) ⊂ H → H the adjoint operator of A. Recall that the domains D(A) and D(A∗ ) are dense in H, see, e.g., [41, Satz VII.4.6, p. 351]. Let (Ω, F, (Ft )t≥0 , P) be a filtered probability space satisfying the usual conditions. Let U be another separable Hilbert space. Whenever there is no ambiguity possible, we also denote by h·, ·i its inner product, and by k · k its associated norm. Let Q ∈ L(U ) be a compact, self-adjoint, strictly positive linear operator. Then there exist an orthonormal basis {ej } of U and a bounded sequence λj of strictly positive real numbers such that X Qu = λj hu, ej iej , u ∈ U j

namely, the λj are the eigenvalues of Q, and each ej is an eigenvector corresponding to λj , see, e.g., [41, Thm. VI.3.2].

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

26

1

1

1

The space U0 := Q 2 (U ), equipped withpinner product hu, viU0 := hQ− 2 u, Q− 2 viU , is another separable Hilbert space and { λj ej } is an orthonormal basis. P Let W be a Q-Wiener process [10, p. 86,87]. We assume that Tr Q = j λj < ∞. Otherwise, which is the case if W is a cylindrical Wiener process, there always exists a separable Hilbert space U1 ⊃ U on which W has a realization as a finite trace class Wiener process, see [10, Chap. 4.3]. We denote by L02 (H) := L2 (U0 , H) the space of Hilbert-Schmidt operators from U0 into H, which, endowed with the Hilbert-Schmidt norm sX 0 λj kΦej k2 , Φ ∈ L02 (H) kΦkL2 (H) := j

itself is a separable Hilbert space. According to [10, Prop. 4.1], the sequence of stochastic processes {β j } defined as j β := √1 hW, ej i is a sequence of real-valued independent (Ft )-Brownian motions λj

and we have the expansion (A.1)

W =

Xp

λj β j ej ,

j

where the series is convergent in the space M 2 (U ) of U -valued square-integrable martingales. Let Φ : Ω × R+ → L02 (H) be an integrable process, i.e. Φ is predictable and satisfies Z T  P kΦt k2L0 (H) dt < ∞ = 1 for all T ∈ R+ . 2

0

Setting Φj :=

p

(A.2)

λj Φej for each j, we have Z t XZ t Φs dWs = Φjs dβsj , 0

j

t ∈ R+

0

where the convergence is uniformly on compact time intervals in probability, see [10, Thm. 4.3]. Let (E, E) be a measurable space which we assume to be a Blackwell space (see [11, 22]). We remark that every Polish space with its Borel σ-field is a Blackwell space. Furthermore, let µ be a homogeneous Poisson random measure on R+ × E, see [25, Def. II.1.20]. Then its compensator is of the form dt ⊗ F (dx), where F is a σ-finite measure on (E, E). We shall now focus on (semi-linear) stochastic partial differential equations (A.3) ( drt

=

r0

(Art + α(rt ))dt +

P

j

σ j (rt )dβtj +

R E

γ(rt− , x)(µ(dt, dx) − F (dx)dt)

= h0

on the separable Hilbert space H with coefficients α : H → H, σ : Hp → L02 (H) and j j γ : H × E → H. In (A.3), we have defined σ : H → H as σ (h) := λj σ(h)ej for each j. The initial condition is an F0 -measurable random variable h0 : Ω → H. A.1. Definition. An adapted, c` adl` ag H-valued process (rt )t≥0 is called a strong solution for (A.3) with r0 = h0 if we have rt ∈ D(A), t ≥ 0, the relation Z t   Z 2 2 P kArs + α(rs )k + kσ(rs )kL0 (H) + kγ(rs , x)k F (dx) ds < ∞ = 1 0

2

E

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

27

for all t ∈ R+ , and t

Z

XZ

(Ars + α(rs ))ds +

rt = h0 + 0

j

t

σ j (rs )dβsj

0

Z tZ γ(rs− , x)(µ(ds, dx) − F (dx)ds),

+ 0

t ≥ 0.

E

A.2. Definition. An adapted, c` adl` ag H-valued process (rt )t≥0 is called a weak solution for (A.3) with r0 = h0 if Z t    Z 2 2 kα(rs )k + kσ(rs )kL0 (H) + P kγ(rs , x)k F (dx) ds < ∞ = 1 (A.4) 2

0

E

for all t ∈ R+ , and for all ζ ∈ D(A ) we have Z t XZ t hζ, rt i = hζ, h0 i + (hA∗ ζ, rs i + hζ, α(rs )i)ds + hζ, σ j (rs )idβsj ∗

0

j

0

Z tZ hζ, γ(rs− , x)i(µ(ds, dx) − F (dx)ds),

+

t ≥ 0.

E

0

A.3. Definition. An adapted, c` adl` ag H-valued process (rt )t≥0 is called a mild solution for (A.3) with r0 = h0 if we have (A.4) for all t ∈ R+ , and Z t XZ t rt = St h0 + St−s α(rs )ds + St−s σ j (rs )dβsj 0

0

j

Z tZ St−s γ(rs− , x)(µ(ds, dx) − F (dx)ds),

+ 0

t ≥ 0.

E

By convention, uniqueness of solutions for (A.3) T is meant up to indistinguishability, that is, for two solutions r1 , r2 we have P( t∈R+ {rt1 = rt2 }) = 1. A.4. Assumption. There exist another separable Hilbert space H, a C0 -group (Ut )t∈R on H and continuous linear operators ` ∈ L(H, H), π ∈ L(H, H) such that the diagram U

t H −−−− → x  `

H  π y

S

t H −−−− → H commutes for every t ∈ R+ , that is

πUt `h = St h for all t ∈ R+ and h ∈ H. R A.5. Assumption. We assume E kγ(0, x)k2 F (dx) < ∞. A.6. Assumption. We assume there is a constant L > 0 such that kα(h1 ) − α(h2 )k ≤ Lkh1 − h2 k, Z E

for all h1 , h2 ∈ H.

kσ(h1 ) − σ(h2 )kL02 (H) ≤ Lkh1 − h2 k,  21 kγ(h1 , x) − γ(h2 , x)k2 F (dx) ≤ Lkh1 − h2 k

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

28

A.7. Theorem. [20, Thm. 8.6, Cor. 10.6] Suppose that Assumptions A.4, A.5, A.6 are fulfilled. Then, for each h0 ∈ L2 (Ω, F0 , P; H) there exists a unique c` adl` ag, adapted, mean square continuous mild and weak H-valued solution (ft )t≥0 for X dft = U−t `α(πUt ft )dt + U−t `σ j (πUt ft )dβtj j

(A.5)

Z U−t `γ(πUt ft− , x)(µ(dt, dx) − F (dx)dt)

+ E

with f0 = `h0 satisfying   E sup kft k2 < ∞

for all T ∈ R+ ,

t∈[0,T ]

and there exists a unique c` adl` ag, adapted, mean square continuous mild and weak H-valued solution (rt )t≥0 for (A.3) with r0 = h0 satisfying   2 E sup krt k < ∞ for all T ∈ R+ , t∈[0,T ]

which is given by rt := πUt ft , t ≥ 0. A.8. Remark. By a solution (ft )t≥0 for (A.5) with f0 = `h0 we precisely mean that Z t P kU−s `α(πUs fs )k + kU−s `σ(πUs fs )k2L0 (H) 2 0   Z + kU−s `γ(πUs fs , x)k2 F (dx) ds < ∞ = 1 E

for all t ∈ R+ , and we have Z t XZ t ft = `h0 + U−s `α(πUs fs )ds + U−s `σ j (πUs fs )dβsj 0

0

j

Z tZ U−s `γ(πUs fs− , x)(µ(ds, dx) − F (dx)ds),

+ 0

t ≥ 0.

E

˜ t )t≥0 A.9. Lemma. Let τ be a bounded stopping time. We define the new filtration (F ˜ ˜ ˜ by Ft := Fτ +t , the new U -valued process W by Wt := Wτ +t − Wτ and the new random measure µ ˜ on R+ × E by µ ˜(ω; B) := µ(ω; Bτ (ω) ), B ∈ B(R+ ) ⊗ E, where Bτ := {(t + τ, x) ∈ R+ × E : (t, x) ∈ B}. ˜ t )t≥0 and µ ˜ is a Q-Wiener process with respect to (F Then W ˜ is a homogeneous ˜ t )t≥0 having the compensator Poisson random measure on R+ × E with respect to (F dt ⊗ F (dx). Moreover, we have the expansion Xp ˜ = (A.6) λj β˜j ej , W j

˜ t )where β˜j defined as β˜tj := βτj +t − βτj is a sequence of real-valued independent (F Brownian motions. Furthermore, if (rt )t≥0 is a weak solution for (A.3), then the ˜ t )-adapted process (˜ (F rt )t≥0 defined by r˜t := rτ +t is a weak solution for (A.7) ( d˜ rt r˜0

=

(A˜ rt + α(˜ rt ))dt +

= rτ .

P

j

σ j (˜ rt )dβ˜tj +

R E

γ(˜ rt− , x)(˜ µ(dt, dx) − F (dx)dt)

TERM STRUCTURE MODELS DRIVEN BY POISSON MEASURES

29

˜ t )-adapted process with W ˜ is a continuous (F ˜ 0 = 0, and µ Proof. Note that W ˜ is an integer-valued random measure on R+ × E. We fix u ∈ U . The process Mt :=

exp(ihu, Wt i) , E[exp(ihu, Wt i)]

t≥0

is a complex-valued martingale, because for all s, t ∈ R+ with s < t the random variable Wt − Ws and the σ-algebra Fs are independent. The martingale (Mt )t≥0 admits the representation   t Mt = exp ihu, Wt i + hQu, ui , t ≥ 0. 2 According to the Optional Stopping Theorem, the process (Mt+τ )t≥0 is a nowhere ˜ t )-martingale. Thus, for s, t ∈ R+ with s < t we obtain vanishing complex (F   Mt+τ ˜ E | Fs = 1. Ms+τ ˜ s we get For each C ∈ F ˜t − W ˜ s i)] = P(C) exp E[1C exp(ihu, W



 t−s hQu, ui . − 2

˜ s are independent, and W ˜ t −W ˜ s and the σ-algebra F ˜ t− Hence, the random variable W ˜ Ws has a Gaussian distribution with covariance operator (t − s)Q. The expansion (A.6) follows from (A.1). Now fix v ∈ R and B ∈ E with F (B) < ∞. The process Nt :=

exp(ivµ([0, t] × B)) , E[exp(ivµ([0, t] × B))]

t≥0

is a complex-valued martingale, because for all s, t ∈ R+ with s < t the random variable µ((s, t] × B) and the σ-algebra Fs are independent. By [25, Thm. II.4.8] the martingale (Nt )t≥0 admits the representation   Nt = exp ivµ([0, t] × B) − (eiv − 1)F (B)t , t ≥ 0. According to the Optional Stopping Theorem, the process (Nt+τ )t≥0 is a nowhere ˜ t )-martingale. Thus, for s, t ∈ R+ with s < t we obtain vanishing complex (F   Nt+τ ˜ E | Fs = 1. Ns+τ ˜ s we get For each C ∈ F   E[1C exp(iv µ ˜((s, t] × B))] = P(C) exp (eiv − 1)F (B)(t − s) . ˜ s are independent, and Hence, the random variable µ ˜((s, t] × B) and the σ-algebra F µ ˜((s, t] × B) has a Poisson distribution with mean (t − s)F (B). Next, we claim that Z τ +t Z t ˜s (A.8) Φs dWs = Φτ +s dW τ

0

for every predictable process Φ : Ω × R+ → L02 (H) satisfying Z t  2 P kΦs kL0 (H) ds < ∞ = 1 0

2

30

´ STEFAN TAPPE, AND JOSEF TEICHMANN DAMIR FILIPOVIC,

for all t ∈ R+ , and Z (A.9)

τ +t

Z Ψ(s, x)µ(ds, dx) =

τ

t

Ψ(τ + s, x)˜ µ(ds, dx) 0

for every predictable process Ψ : Ω × R+ × E → L02 (H) satisfying  Z tZ 2 kΨ(s, x)k F (dx)ds < ∞ = 1 P 0

E

for all t ∈ R+ . If Φ, Ψ are elementary and τ a simple stopping time, then (A.8), (A.9) hold by inspection. The general case follows by localization. If (rt )t≥0 is a weak solution to (A.3), for every ζ ∈ D(A∗ ) relations (A.8), (A.9) yield Z τ +t X Z τ +t hζ, σ(rs )idβsj hζ, rτ +t i = hζ, rτ i + (hA∗ ζ, rs i + hζ, α(rs )i)ds + τ

Z

τ

j

τ +t Z

hζ, γ(rs− , x)i(µ(ds, dx) − F (dx)ds)

+ τ

E

Z = hζ, rτ i +

t ∗

(hA ζ, rτ +s i + hζ, α(rτ +s )i)ds + 0

XZ j

t

hζ, σ(rτ +s )idβ˜sj

0

Z tZ hζ, γ(r(τ +s)− , x)i(˜ µ(ds, dx) − F (dx)ds).

+ 0

E

Hence, (˜ rt )t≥0 is a weak solution for (A.7).



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